Math
Rørdam K-theory
2.2.10 Liftings
Chapter 2 Projections and Unitary elements
Chapter 3 The K_0 Group of a Unital C -algebra.
Chapter 4 The functor K0
Chapter 5 The ordered abelian group K0
Chapter 6 Inductive limits
Characterization for stable equivalence
Corollary 2.1.4
Definition 2.1.2.
Definition 2.2.1
Definition 2.3.1 (The semigroup of projections of A)
Definition 2.3.3. (The semigroup of projections)
Definition 3.1.4 (The K_0 group of a Unital C-algebra)
Definition 3.1.6 (Stable equivalence)
Definition 3.2.5 (Homotopy equivalence)
Definition 4.1.1 (K0, non-unital case)
Definition 4.2.1 (The Scalar mapping)
Definition 5.1.1.
Definition 5.1.3
Definition 5.1.4 (Positive cone of K0)
Definition 5.16
Definition 6.4.a (Stabilization)
Drawing 2024-02-05 22.07.46
DRAWING
Example 3.1.3
Example 3.3.2
Example 3.3.3
Example 3.3.5
Example 4.1.5
Example 4.3.5
Lemma 2.1.3
Lemma 2.1.5 (Whitehead)
Lemma 2.1.7.
Lemma 2.2.3
Lemma 3.2.7
Lemma 3.2.8
Lemma 4.3.2
Lemma 5.1.2
Lemma 6.3.1
Murray-von Neumann equivalence
Paragraph 3.3.1 Traces and K0
Properties of the Grothendieck group
Proposition 2.1.6.
Proposition 2.1.8
Proposition 2.1.11
Proposition 2.2.5
Proposition 2.2.6
Proposition 2.2.7
Proposition 2.2.8
Proposition 2.3.2.
Proposition 3.1.7 (The standard picture for K0 — the unital case)
Proposition 3.1.8 (Universal property of K0)
Proposition 3.2.4 (Functoriality of K0 for unital C-star Algebras)
Proposition 3.2.6 (Homotopy invariance of K0)
Proposition 4.1.3 (Functoriality of K0)
Proposition 4.1.4 (Homotopy invariance of K0)
Proposition 4.3.2 (Half exactness of K0)
Proposition 4.3.3 (Split exactness of K0)
Proposition 4.3.4 (Direct sums)
Proposition 4.3.8 (Stability)
Proposition 5.1.5
Proposition 5.1.7
Proposition 5.1.9.
Proposition 6.2.4
Proposition 6.4.1 (Stability of K0)
retract
Rørdam K-theory
The functor K00 3.2.3
The functor K0 for unital C-star Algebras 3.2.2
The Grothendieck construction 3.1.1
The group K_00 3.1.5
Theorem 2.1.9. (The polar decomposition)
Theorem 2.1.10 (Carl Neumann Series)
Theorem 4.1.2. (The functor K0)
Theorem 5.2.1
Theorem 5.2.2
Theorem 6.3.2 (Continuity of K0)
Diffuse measure
dirac measures
finite C-star algebra
finite projection
homotopy
infinite C-star algebra
infinite projection
probability measure
quasi-trace
stably finite

Chapter 4 The functor K0

We extend the functor to non-unital -algebras and make sure that the new extension agrees with the old definition given for unital -algebras.
Then we show that the functor is split exact, half-exact, homotopy invariant and stable under amplifications.

Definition 4.1.1 (K0, non-unital case)
Let be a non-unital -algebra, and consider the split exact sequence associated to the Unitizations of .
Define

Since the functor is split exact, we have that no matter whether the algebra is unital or not, the short exact sequence which arises from the unitization, induces an exact sequence of the k-theory groups.

The map is when is unital, and if it is not unital, then it is the inclusion map.

Theorem 4.1.2. (The functor K0)
Let be a -homomorphism, let be its unitization. The commutative diagram

The induced diagram has a commuting square on the right, since is functoriality for unital -algebras, and then in order to get a commuting square on the left, must be defined to be the restriction of to .

Notice that

  1. for any -algebra.
  2. If and are -algebras, and if and then
  1. If are homotopic -homomorphisms, then
  2. If are homotopy equivalent -algebras, then is isomorphic to . More specifically, if is a homotopy, then are isomorphisms and .

Example 4.1.5
The cone of an algebra and the suspension of an algebra are defined as We now have a short exact sequenceThe first map is the embedding, the second is the the evaluation of at .

The cone of an algebra is homotopy equivalent to the zero -algebra. Thus .

4.2 The standard picture of the group K0

Proposition 4.2.2 and Lemma 4.2.3 below contain the standard picture of a handy description of what elements in look like. These results will be invoked whenever explicit calculations involving - groups are performed.

Definition 4.2.1 (The Scalar mapping)
Consider the split exact sequence obtained by adjoining a unit to the -algebra . Define the scalar mapping to be In other words, for all and . Notice that , and that belongs to for each . Let be the star homomorphism induced by . The image of is the subset of consisting of all matrices with scalar entries, and again for all elements in the unitzation. An element in or will be called a scalar element if .
The scalar mapping is natural in the sense that if and are -algebras, and if is a star homomorphism, then we get a commuting diagram, where .

Proposition 4.2.2. (The standard picture of General case)
One has that for each -algebra , that Moreover, the following hold.

  1. For each pair of projections the following conditions are equivalent:
    (a).
    (b). there exist natural numbers such that , in other words, stable equivalence.
    (c). there exist scalar projections and such that
  2. If satisfies , then there is a natural number with .
  3. If is a star homomorphism , then
    for all projections in .

Lemma 4.2.2
Let and be -algebras, and let be a -homomorphism. Suppose that belongs to the kernel of .

  1. There exist a natural number , a projection , and a unitary in such that and
    In other words, if is in the kernel of the induced K0 map, then its form in the k theory of B is standard with a projection “p”, and the image of p under the unitized map is unitarily equivalent to its own scalar part

  2. If is surjective, then there is a projection , , and . In other words, the unitary is just 1!.

4.3 Half and split exactness and stability of K0

We begin with a description of what happens when units are adjoined to a short exact sequence. The straightforward proof is left to us.

Lemma 4.3.1.
Let be a short exact sequence of -algebras, and let .

  1. The mapping is invective.
  2. An element a in belongs to the image of if and only if

Proposition 4.3.2 (Half exactness of K0)
Every short exact sequence of -algebras, induces an exact sequence of Abelian groups

Proposition 4.3.3 (Split exactness of K0)
Every split exact sequence of -algebras induces a split exact sequence of K-theories.
Proposition 4.3.4 (Direct sums)
K-theory respects direct sums.
For any pair of -algebras we have that Proof: five lemma.

Example 4.3.5
From the split exactness, and traces, we conclude that

Proposition 4.3.8 (Stability)
Let be a -algebra and let . Then .

Table Of Contents
Chapter 4 The functor K0
4.2 The standard picture of the group K0
4.3 Half and split exactness and stability of K0